Professor Stewart's Hoard of Mathematical Treasures (35 page)

The question now remains: which whole numbers d can be the area of a Pythagorean triangle with rational sides? The answer is not obvious. It turns out to be linked to a different equation,
which has solutions p, q in whole numbers if and only if d is congruent.

Some numbers are congruent, some aren’t. For example, 5, 6, 7 are congruent, but 1, 2, 3, 4 aren’t. It need not be straightforward to decide: for example, 157 is a congruent number, but the simplest right triangle with area 157 has hypotenuse
The best test currently known depends on an unproved conjecture, the Birch-Swinnerton-Dyer conjecture, which is one of the Clay millennium mathematics prizes (Cabinet, page 127) with a million dollars on offer for a proof or disproof. Frederick II didn’t realise what he was starting.

Norbert Wiener pioneered the mathematics of random processes, as well as the new subject of cybernetics, in the first half of the 20th century. He was a brilliant mathematician, and notorious for forgetting things. So when the family moved to a new house, his wife wrote the address on a slip of paper and gave it to him. ‘Don’t be silly, I’m not going to forget anything as important as that,’ he said, but he put the paper in his pocket anyway.

Norbert Wiener.

Later that day, Wiener became immersed in a mathematical problem, needed some paper to write on, took out the slip bearing his new address, and covered it in equations. When he had finished these rough calculations, he crumpled the paper into a ball and threw it away.

As evening approached, he recalled something about a new house but couldn’t find the slip of paper with its address. Unable to think of anything else to do, he walked to his old house, and noticed a little girl sitting outside it.

‘Pardon me, my dear, but do you happen to know where the Wieners have mov—’

‘That’s OK, Daddy. Mommy sent me to fetch you.’

Crossnumber grid.

A crossnumber is like a crossword, but using numbers instead of words. All the clues for this one are about time, and are prefaced by the phrase ‘the number of ...’.

Answer on page 319

If all these statements are correct, do I avoid kangaroos, or not?

 

In the late 1800s, there was a vogue for naming special surfaces after mathematicians: Kummer’s surface, for instance, was named after Ernst Eduard Kummer. The mathematicians tended to be German, and the German word for surface is Fläche, so this was the ‘Kummersche Fläche’. I’m delving into the linguistics here, because it led to a pun being used to name a mathematical concept. That still happens, but this may well have been the first occasion. The pun derives from a very similar word, Flasche, which means ‘bottle’. At any rate, the scene was set: when Felix Klein invented a bottle-shaped surface in 1882, it was naturally called the ‘Kleinsche Fläche’. And inevitably this rapidly mutated into ‘Kleinsche Flasche’ - the Klein bottle.

I don’t know whether the pun was deliberate, or a mistranslation. At any rate, the new name was so successful that even the Germans adopted it.

Klein’s surface . . .

... interpreted as a bottle.

The Klein bottle is important in topology, as an example of a surface with no edges and only one side. A conventional surface, such as the sphere - by which topologists just mean the thin skin of the sphere’s surface, and not a solid ball (which they call a ball) - has two distinct sides, an inside and an outside. You can imagine painting the inside red and the outside blue, and the two colours never meet. But you can’t do that with a Klein bottle. If you start painting what looks like the outside blue, you get to the bent tube where it becomes narrower, and if you follow that tube as it penetrates through the bulging body you end up painting what looks like the inside blue as well.

Klein invented his bottle for a reason: it came up naturally in the theory of Riemann surfaces in complex analysis, which classifies nasty kinds of behaviour - in a beautiful way - when you try to develop calculus over the complex numbers. The Klein bottle is reminiscent of an even more famous surface, the Möbius band (or strip), formed by twisting a strip of paper and gluing the ends together. The Möbius band has one side, but it also has an edge (Cabinet, page 111). The Klein bottle gets rid of the edge, which topologists find more convenient because edges can cause trouble. Especially in complex analysis.

There is a price to pay, however: the Klein bottle can’t be represented in ordinary 3D space without penetrating through itself. However, topologists don’t mind that, because they don’t
represent their surfaces in 3D space anyway. They prefer to think of them as abstract forms in their own right, not relying on the existence of a surrounding space. In fact, you can fit a Klein bottle into 4D space without any interpenetration, but that brings its own difficulties.

One way to represent a Klein bottle, which doesn’t require any self-intersection, is to borrow a trick that is familiar to almost everyone nowadays from computer games. (The topologists thought of it long before, I hasten to add.) In many games, the flat rectangular computer screen is ‘wrapped round’ so that the left and right edges are in effect joined together. If an alien spaceship shoots off the right-hand edge, it immediately reappears at the left-hand edge. The top and bottom may also be wrapped round in this manner. Now, a computer screen doesn’t actually bend. Much. So the ‘wrapping round’ is purely conceptual, a figment of the programmer’s mind. But we can easily imagine that the opposite edges abut, work out what would happen if they did, and respond accordingly. And that’s what topologists do.

Specifically, they also start with a rectangle, and wrap its edges round so that in the imagination they join. But there’s a twist - literally. The top and bottom edges are wrapped round as usual, but the right edge is given a half-twist, interchanging top and bottom, before wrapping it round to meet the left edge. So when a spaceship shoots off the top, it reappears from the corresponding position at the bottom; but when it shoots off the right-hand edge, it reappears upside down and at the opposite end of the left-hand edge.

Conventional computer screen Klein bottle wrap-round. wrap-round.

Topologically, the conventional wrapped-round screen is a torus - like a car inner tube or (I have to say this because many people have never seen an inner tube, since most car tyres are tubeless) a doughnut. But only the sugary surface, not the actual dough. You can see why if you imagine what happens when you do actually join the edges - using a flexible screen. Joining top to bottom creates a cylindrical tube; then joining the ends of the tube bends it round into a closed loop.